RELATIVITY
Frames of Reference
Special Relativity
The time required for the light beam to travel from the splitter to the mirror is obtained from . This still implies that the "laws" of electromagnetism behave differently during a transformation from one frame of reference to another than the "laws" of mechanics. It doesn't seem reasonable that one "part" of physics should be different from another "part" of physics.
He postulated that 'All the "laws" of physics are the same in all inertial reference frames.'. The speed of light in a vacuum is (measured as) the same in all inertial reference frames.'. When the speed of light is measured in the two reference frames, it is found that c≠c′+v, rather c=c′.
Consequences of the Principle of Special Relativity
So, for example, in the situation discussed above, the width and thickness of the meter stick are still measured the same in both reference frames. The proper length of an object is that length measured in the remaining frame of the object. Flashes of light are emitted at points C1 and C2 when the origins (O & O') of the two frames coincide.
Nevertheless, both observers measure the same speed of light, approx. Now we want to derive the transformation equations for an object's displacement and velocity - the relativistic version of the Galilean transformation equations. Subtract the second term from the first and collect the S-frame on one side of the equals sign, the S-frame on the other side. This is an extension of the invariance of lengths under a rotation of the coordinate axes.
Energy and Momentum
An electron-volt is the energy gained by an electron when it is accelerated through a potential difference of one volt. Remember that Eo is the relativistic total energy of the second sphere, while mc2 is the rest energy of the first (target) sphere. Knowing the momenta and masses of the decay products, we determine the mass of the incident particle in the hope of identifying it. c p1 =910 MeV.
Now that we have the total energy and the kinetic energy, the mass is taken from. When we talk about total stored energy that includes total resting energy. The rest energy of the anti-neutrino is too small to worry about. ii) Keep in mind the rounding of numbers and significant figures when substituting numerical values in formulas. iii) Note that mn ≠mp +me.
A Hint of General Relativity
QUANTUM THEORY
The radiation is said to be in thermal equilibrium with the furnace walls. From thermodynamics (Kirchhoff), the effect radiated by a body in thermal equilibrium is expressed by radiation. A physical model for an ideal black body is a small opening in the wall of a heated cavity.
Because the aperture is small, a ray of light entering through the aperture is very unlikely to bounce back. Conversely, any rays of light that emerge through the opening will have reached equilibrium with the interior walls, bouncing off the walls many times over. The black body is not the furnace as a whole, but the opening in the wall of the furnace.
Black Body Radiation
Rayliegh proposed that the energy density be expressed as the product of the number of modes of a standing wave in a cavity and the average energy of each mode. Classically, the probability that an energy mode will exist in the cavity, E, is given by a Boltzmann distribution. Because we have to have standing waves in the box, with the electric field disappearing on the walls,.
In other words, only discrete points in k-space define the allowed energy modes in the cubic cavity. So we count the number of points k that lie in a spherical shell of radius k and thickness dk. Finally, there are two perpendicular polarizations for each mode, so the number of modes per unit volume (density of states) is.
Compared to the observed blackbody spectrum, the Rayleigh-Jeans "law" differs as λ→0. The range of allowed energies for standing waves in the cavity was treated as a continuous variable. Instead of visualizing standing E-M waves inside a cavity, consider the atoms that make up the walls of the cavity.
These atoms vibrate and absorb or emit E-M waves.. energies of these oscillators can only change in discrete steps, rather than continuously. This result is multiplied by the number of modes of frequency f, 3 8 2 .. i) Assume that oscillators or standing waves are limited to discrete values of energy, En = nfh. ii). therefore, the probability that a high-frequency or short-wavelength mode is occupied or present is very low. iii). An oscillator that emits energy can only change its energy in steps of ∆E=nhf , where f is the frequency of vibration of the oscillator.
Adjusting Planck's formula to the observed blackbody radiation yields a value for Planck's constant, h = 6.626x10-34J sec.
Photons
We conclude that E-M radiation is not a continuous waveform, but consists of discrete, localized wave packets, called a photon. The classic prediction is that the electron will experience acceleration, causing the electron to oscillate transversely and to move in the direction the light wave is traveling. The longer the electron is exposed to the incident radiation, the faster it translates and the greater the Doppler shift.
The intensity of the incident radiation also affects the Doppler shift because it affects the acceleration of the electron. Qualitatively, it is observed that the frequency, f′, of the scattered photon depends only on the scattering angle, θ. It is necessary to use the relativistic forms of energy and momentum, since the photon is certainly moving with a speed close to, if not equal to, c.
Matter Waves
In sound waves, the regular rises and falls in amplitude are known as beats. The individual harmonic waves travel at different speeds, so the wave packet or wave group spreads out over time. We suppose that the motion of the particle can be modeled by a traveling wave packet with frequency fo = E/h and wavelength λo = h/p, where E is the total relativistic energy and p is the total relativistic momentum of the particle.
The wave packet would be constructed by a superposition of harmonic waves with wavelengths centered at λ0. Therefore, the group velocity of the wave packet coincides with the velocity of the massive particle: vg = v. One of the striking features of waves is the fact that they interfere with each other and form interference patterns.
When a monochromatic light beam is incident on a diffraction grating, a characteristic interference pattern is observed, which is fully understood in terms of constructive and destructive interference between the waves of scattered light. We imagine that the rows of atoms that make up the surface of the crystal correspond to closely spaced lattice lines.] We could solve for the wavelength: λ = d·sin(ϕmax). For two slits spaced apart by a distance D and only one open in succession, a superposition of single-slit distributions would be observed.
The wave function is interpreted as related to the probability that a measurement will find a particle at a particular location, x'. The wave function is non-zero for a region on the x-axis of width ∆x centered at x = 0. The total probability of a particle being observed somewhere on the x-axis must be 1.0, so it is a wave function.
The height of the central peak is much greater than the side peaks, and the width of the central peak is approx.
Atoms
QUANTUM MECHANICS & ATOMIC STRUCTURE (ABBREVIATED)
Schrödinger Wave Equation—One Dimensional
The probability that a particle will be observed at a location between x and x+dx at time t is given by P(x,t)dx= Ψ(x,t)2dx, where Ψ2 =Ψ*Ψ and P(x ,t ) is the probability density. Later we will have to explain the connection between Ψ2 and the physical motion of the particle. Solving this equation can be easy or difficult depending on the form of U(x).
One-Dimensional Potentials
1, where pi is the number of times the value Ai appears in the list of N different values. Now Ψ2 is the probability density, which gives the odds that the particle will be observed to be in the state described by Ψ(x,t).
The Hydrogen Atom
We can calculate the expected value of r, the distance from the electron to the atomic nucleus. Note: Although r = ao is the most likely radius, the expectation value of the radius is
On the other hand, the magnitude of the orbital angular momentum is according to Kepler. In the presence of a strong magnetic field, the interaction of the orbital magnetic dipole with the external magnetic field results in a change in energy. with The normal Zeeman effect is therefore characterized by the splitting of spectral lines into sets of several evenly spaced lines in the presence of a strong magnetic field.
The observed result is that the atomic beam is split into two beams with equal but opposite deflections. Interpretation: i) the effect is not due to the orbital magnetic moment, otherwise there would be either 2+1 or zero spots on the screen and ii) the deflections are due to a magnetic moment that has only two possible values. By analogy with orbital angular momentum, we define a spin quantum number, s, such that the number of possible angular momentum z-components is 2s+1.
This S2 is constant, so this thing called spin is as much an intrinsic property of the electron as its mass and electric charge. This value indicates that the electric charge is not evenly distributed in a small, solid rotating sphere, as our classic visualization shows. These two contributions determine the total magnetic moment of an electron orbiting a proton.
The uses of the 2 schemes of spectroscopic notation are as follows: for orbital subshells we use the lowercase letters s, p, d, f, etc.; for the energy states we use the capital letters S, P, D, etc.
Multi-electron Atoms
